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Let $$f\colon X \to Y$$ be a morphism of affine normal algebraic varieties over $\mathbb{C}$. Assume that $f$ is birational and bijective on closed points. Does normality imply that $f$ is an isomorphism? Does it follow from Stein's factorization?

(I am especially interested in the surface case, but I do not see why this should help.)

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    $\begingroup$ There is also the related notion of seminormality. $Y$ (a variety over $\mathbb{C}$) is called seminormal if every map of varieties $X \to Y$ which is finite and bijective is an isomorphism (and hence birational as well). Normal implies seminormal. $\endgroup$
    – Karl Schwede
    Commented Jun 19, 2015 at 16:57
  • $\begingroup$ Thanks. Any reference about semi-normality? Or criteria to test it? I have never come across it. $\endgroup$
    – Giulio
    Commented Jun 21, 2015 at 13:48
  • $\begingroup$ I think probably googling it will give you some references. The classic reference is Greco-Traverso. There was a recent survey (from an algebraic perspective) by Marie Vitulli. $\endgroup$
    – Karl Schwede
    Commented Jun 22, 2015 at 5:30

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Please confer Zariski's Main Theorem, pp. 288-289 of Mumford's "Red Book of Varieties and Schemes". Using the "original form", $f$ is an open immersion. Using the hypothesis about bijectivity on closed points, $f$ is an isomorphism.

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  • $\begingroup$ Thanks for the reference. In Section III.9 Page 209 Second Edition he addresses explicitly my question. $\endgroup$
    – Giulio
    Commented Jun 19, 2015 at 13:25

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